Constructing a Position-Time Function for a Rocket

[Nabeshin]

Suppose that I want to launch a rocket and want to construct a position-time function for the rocket so that I know where it will be at any given time t. To make this realistic, say mass varies with time, so as the rocket burns fuel it becomes lighter. Suppose the rocket, burning fuel at a constant rate, provides a constant force to the rocket. Now, in trying to construct a position vs. time function, I have to consider gravity.

I know $$F_g=\frac{Gm_1m_2}{r^2}$$, so the force of gravity varies with respect to position. And, of course, the position is a function of time. So I find a strange recursion here, and I know that I am thinking about this problem in the wrong way.

Perhaps I need to think more in terms of momentum or energy?

 

[arildno]

Suppose that I want to launch a rocket and want to construct a position-time function for the rocket so that I know where it will be at any given time t. To make this realistic, say mass varies with time, so as the rocket burns fuel it becomes lighter. Suppose the rocket, burning fuel at a constant rate, provides a constant force to the rocket. Now, in trying to construct a position vs. time function, I have to consider gravity.
I know $$F_g=\frac{Gm_1m_2}{r^2}$$, so the force of gravity varies with respect to position. And, of course, the position is a function of time. So I find a strange recursion here, and I know that I am thinking about this problem in the wrong way.

Not at all! To solve such "strange recursion" problems, as you call them, or "implicit equations" as mathematicians might call them, is the very core of the branch called "applied mathematics".
In your case, you'll need to solve a specific form of this type of recursion, namely what we call a differential equation.

 

[Nabeshin]

Ok. I have only a very modest background in differential equations, what was glossed over in calc I, so this is (most likely) beyond the scope of what I know. But if I consider the sum of the forces and F=ma, I get something like this:

$$\frac{d^2r}{dt^2}=\frac{F_r}{m_r}-\frac{Gm_e}{r^2}$$

Where $$F_r$$ is the force produced by the rocket.. It would probably help, at this point, to write the mass of the rocket as something like:
$$m_r=m_{r0}-R*t$$ where R is the rate at which mass is lost due to fuel consumption.

$$\frac{d^2r}{dt^2}=\frac{F_r}{m_{r0}-R*t}-\frac{Gm_e}{r^2}$$

As I said my only differential equation background is from Calc I so really the only thing I can try to think of to do is to split the r's and t's and integrate, but I can't manage to do that :( :(

 

[rcgldr]

It might be possible to integrate this, but I don't know how. Often times, problems like this require numerical intergration using equations to caculate accelerations, and then a method like Runge-Kutta to predict new velocities and positions, then repeat the acceleration calculations step by step over small time intervals to predict a path.

 

[Nabeshin]

So basically you're saying that this may not have an explicit solution and that it will have to be approximated?

 

[rcgldr]

So basically you're saying that this may not have an explicit solution and that it will have to be approximated?

Yes, take a look at this thread in just dealing with gravity in 2 dimensions:

https://www.physicsforums.com/showthread.php?t=264702

Numerical integration has been good enough for space travel to the moon and other planets, although some in flight correction is usually needed to keep spacecraft on the desired path. I'm not sure if this is because inaccuracies in the numerical intergration or because relativistic effects are typically ignored and just compensated for in flight with those minor corrections.

 

[Lojzek]

I think rocket's motion should be integrated numerically. Especially if one also considers air resistance and nonvertical start (vertical start is not very economic, unless the rocket's propulsion force is much greater than the force of gravity). If you have any experience in programming, you can easily do the numerical calculation with Euler's method:
it uses aproximation f(x+dx)=f(x)+f'(x)dx to calculate the result at x+dx from a known value at x. In your case the equation is second order, so you use: v(t+dt)=v(t)+a(t)*dt ; r(t+dt)=r(t)+v(t)*dt. Choose a short enough dt and do the iteration over the whole t range.

Of course Runge-Kutta is more efficient, but Euler's method is still good enough if you only want to solve one problem.

 

[D H]

Sorry to resurrect an old thread; a very recent thread referenced this older thread.

Yes, take a look at this thread in just dealing with gravity in 2 dimensions:

https://www.physicsforums.com/showthread.php?t=264702

That thread deals with gravity in one spatial dimension, not two.

Numerical integration has been good enough for space travel to the moon and other planets, although some in flight correction is usually needed to keep spacecraft on the desired path. I'm not sure if this is because inaccuracies in the numerical intergration or because relativistic effects are typically ignored and just compensated for in flight with those minor corrections.

Numerical integration is essential for space travel because there is no closed solution form.

Relativistic effects are tiny, tiny perturbations. The reasons inflight corrections are needed include:

• We can't measure gravity. It instead has to be estimated based on the propagated state. Most onboard processors use a fairly simple model of gravity. Modern flight systems use up to an 8x8 spherical harmonic model, for example. The 200x200 GRACE gravity model is deemed by many to be insufficient for precision orbit determination.
• The gravity model is by necessity expressed in a planet-fixed frame. Once free of the planet's surface, the vehicle's state is propagated in some non-rotating frame. This means the planet's rotational state must be known to compute the non-spherical gravitational effects. The errors here are very small, but can still dominate relativistic effects.
• Launch vehicles go through the atmosphere. Attaining two decimal places of accuracy in modeling atmospheric drag is very, very good.
• The thrusters are noisy chaotic devices. 10% thrust variance is not unheard of. Even after careful calibration a 2-5% variance is quite common.
• The thrusters aren't quite pointing where you think they are aimed.
• The gyros and accelerometers used in spacecraft inertial navigation systems are subject to noise, biases, and scale factor errors (amongst other problems).

 

[SystemTheory]

In the attached sketch I have a comment for the OP and a question for D.H.

I'm looking at rocket launch and vertical flight from the surface of the earth. The gravity radius is rE + h, which can be converted to approximate gravity function g(h). I draw a continuous-time (CT) computer diagram with position feedback to find weight force W(h). Drag is a negative feedback force D(v) as a function of velocity. Thrust T is an independent source from the engine as a function of time.

Acceleration:

a = (T - W - D)/m(t) where mass varies as a function of time to generate thrust.

My comment is that the differential equation is viewed as feedback in an analog computer or digital computer model. Both computers solve via continuous or discrete iteration, or what I learned to visualize as system feedback.

My question to D.H., if it is not off-topic, is how to best approximate m(t) if one is given, say, the impulse curve of a model rocket engine such as published at the NAR website?

Technically it should depend on relative exhaust velocity but I've merely been taking the integral to find the impulse I(t), dividing by total impulse Ip, and finding dm = I(t)/Ip*mp to assign a change in mass as propellant gets used. If there a simple yet more accurate way to find m(t)? Feel free to decline answering or open a new thread, since I don't want to hijack the topic.

Edit: diagram corrected to show 1/m(t) in the feedforward multiplier block.

 

[Nabeshin]

Sorry to resurrect an old thread; a very recent thread referenced this older thread...

God, I can't believe this was only less than a year and a half ago. I suppose it's my fault for linking it and letting everyone see how much more ignorant I was back then, haha.

Cheers, D H!

 

[D H]

What you are missing, SystemTheory, is the relation between force and mass depletion. The ratio of force to the mass or weight depletion rate is called "specific impulse". Here "specific" means "divided my mass" (or by weight, if you want to stick with doofy English units).

In metric units, specific impulse is force divided by the mass depletion rate. As force has units of mass*length/time² and mass depletion rate has units of mass/time, the units of this ratio is that of velocity. In English units, specific impulse is force divided by weight depletion rate, where weight is defined as mass * g₀ (g₀ = 9.80665 m/s²). To people [strike]stuck in the stone age[/strike] who use English units, specific impulse has units of seconds. Whether you use metric units and express specific impulse as velocity or use English units and express specific impulse as a time, it doesn't really matter. The conversion from one to another is pretty simple; multiply or divide by g₀.

The reason specific impulse is useful is because the primary determinants in specific impulse are the chemicals used in the rocket and the thruster geometry. Other key factors are altitude (rockets are at their most efficient in a vacuum) and build-up/trail-off. A model rocket cannot be turned on and off and on and off, so build-up/trail-off is a non-issue for model rockets. Model rockets don't go all that high, so back pressure is not a big concern here either.

By the above reasoning, a good first-order approximation is to assume specific impulse is a constant throughout. Those model rocket pages often list total impulse in Newton-seconds and total mass consumed. Dividing total impulse by total mass consumed yields the average specific impulse for that rocket. Now suppose you have a time-dependent force model T(t). The mass consumption rate is simply T(t)/Isp. With that and a thrust profile you can deduce the rocket's trajectory.


Have you taken a look at NASA Glenn's rocket pages? The index to these pages: http://www.grc.nasa.gov/WWW/K-12/rocket/shortr.html. There is a wealth of material in these pages. The mathematical background is intro level calculus or less. It is quite accessible to a bright high school student.

 

[SystemTheory]

DH, thanks for the information. My background is EE but this is a potential educational concept for high school level. I follow your logic and it should enable me to better explain the basic rocket concept while removing a numerical integration from the simulator model.